Offshore Production and Business Cycle Dynamics with Heterogeneous Firms1

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Abstract:

Cross-country variation in production costs encourages the relocation of production facilities to other countries, a process known as offshoring through vertical foreign direct investment. I examine the effect of offshoring on the international transmission of business cycles. Unlike the existing macroeconomic literature, I distinguish between fluctuations in the number of offshoring firms (the extensive margin) and in the value added per offshoring firm (the intensive margin) as separate transmission mechanisms. The firms' decision to produce offshore depends on the firm-specific level of labor productivity, on fluctuations in the relative cost of effective labor, and on the fixed and trade costs of offshoring. The model replicates the procyclical pattern of offshoring and the dynamics along its two margins, which I document using data from U.S. manufacturing and Mexico's maquiladora sectors. O¤shoring enhances the synchronization of business cycles, and dampens the real exchange rate appreciation generated by aggregate productivity di¤erentials across countries.

1 Introduction

Firms often follow strategies that involve the establishment of
production affiliates at foreign locations with relatively lower
production costs, a process known in the international trade
literature as offshoring through vertical foreign direct investment
(FDI).3 Unlike production under horizontal FDI
- which means that foreign affiliates aim to gain market access by
replicating the operations of their parent firms at the location
where final consumption takes place - the type of offshoring that I
model is primarily motivated by cross-country differences in the
cost of effective labor, as in Helpman (1984), as foreign
affiliates add value to goods consumed in the multinationals'
country of origin.4 The number of offshoring firms (the
extensive margin) and the real value added per offshoring firm (the
intensive margin) fluctuate over the business cycle, and thus
affect output, prices and wages in both the parent and the host
countries.5

I document the business cycle features of offshoring motivated
by lower production costs using data from U.S. manufacturing and
Mexico's maquiladora sector.6 Using the number of
maquiladora plants to reflect the extensive margin, I show that the
total value added and the number of plants in Mexico are strongly
procyclical with the U.S. manufacturing output (Figure 3). In
addition, the business cycle dynamics of the maquiladora sector
differ across the total value added and its extensive margin: The
total value added co-moves almost contemporaneously with U.S.
manufacturing, whereas the number of plants lags the expansions and
contractions in U.S. manufacturing by about four quarters, a result
which highlights the inter-temporal dynamics of the extensive
margin of offshoring.

Despite this empirical evidence, the international
macroeconomics literature does not fully capture the business cycle
dynamics of offshoring along its extensive margin. Burstein, Kurz,
and Tesar (2009) examine the role of production sharing in the
transmission of business cycles in a two-country model in which the
location of plants is fixed over time.7 Bergin, Feenstra, and
Hanson (2007) focus on the importance of offshore production in
amplifying the transmission of shocks across countries, in a model
in which the number of offshoring firms adjusts instantly - rather
than gradually over time as in the data - in response to
simultaneous aggregate shocks in the parent and the host
countries.

To address this issue, I introduce the endogenous creation of
offshore production plants as a firm-specific decision, in a
two-country (North and South), dynamic stochastic general
equilibrium framework. The key features of the model include
endogenous firm entry in the parent country and firm heterogeneity
in labor productivity. Firm entry is subject to a sunk cost
reflecting headquarter activities at home. Following entry in the
North, each firm can use either domestic or foreign labor in the
production of a different variety of goods. The use of foreign
labor involves the establishment of an offshore plant, and is
subject to fixed and trade costs every period.8 Since firms are
heterogeneous in productivity, the decision to produce offshore is
firm-specific: Despite the lower cost of effective labor abroad
(i.e. the lower wage relative to aggregate productivity), only the
more productive firms can afford the fixed and trade costs of
offshoring. The cross-country asymmetry in the cost of effective
labor also implies that offshoring takes place one-way; only some
of the Northern firms have an incentive to produce offshore,
whereas all Southern firms produce at home.

The key results of the paper are as follows. First, the model
generates a procyclical pattern of offshoring that is consistent
with the data from U.S. manufacturing and Mexico's maquiladora
sector. In the model, the number of offshoring firms depends on the
fluctuations in the relative cost of effective labor across
countries. A positive shock to aggregate productivity in the North
encourages domestic firm entry, and causes the domestic wage to
rise above aggregate productivity as labor demand increases to
accommodate firm entry requirements. Notably, the increase in the
cost of effective labor in the North is gradual (because the number
of firms is a stock variable), and causes a gradual increase in the
number of offshoring firms (the extensive margin), as in the data.
Second, offshoring enhances the co-movement of output across
countries. The increase of output in the North (generated by a
country-specific shock to aggregate productivity) and also in the
South (caused by the immediate jump in Northern demand for
offshored varieties, and also by the subsequent relocation of
production by the Northern firms to the South) enhance the
co-movement of output across the two economies.9 The result is
consistent with the empirical regularity documented in Burstein,
Kurz, and Tesar (2009) that country pairs with larger shares of
offshoring-related bilateral trade exhibit larger correlations of
manufacturing output. Third, offshoring narrows price dispersion
across countries, as it reduces the appreciation of the real
exchange rate that follows a domestic increase in aggregate
productivity in the framework with firm entry and endogenously
traded varieties. Thus, offshoring dampens the
Harrod-Balassa-Samuelson effect through a number of channels,
including the transfer of upward pressure from the domestic to the
foreign wage, the reduced size of the domestic non-traded sector,
and the decline in import prices that occurs as offshoring crowds
out the less productive foreign exporters.

This paper is related to a growing body of macroeconomic
literature that focuses on endogenous firm entry and adjustments
along the extensive margin of exports (but not of
offshoring).10 For example, Ghironi and Melitz
(2005) study the export decision of firms in the presence of fixed
exporting costs, in a framework with firm entry and firm
heterogeneity. Alessandria and Choi (2007) analyze the extensive
margin of exports in a model with sunk costs and continuation fixed
costs that explains the "exporter hysteresis" behavior.11
Corsetti, Martin, and Pesenti (2007) examine the terms-of-trade
implications of productivity improvements in the sectors of firm
entry and production, in a model in which the extensive margin of
exports is endogenous. Finally, Mejean (2006) studies the effect of
endogenous firm entry in the tradable sector on the real exchange
rate dynamics and the Harrod-Balassa-Samuelson effect.

The assumptions of the model considered in this paper are
consistent with the empirical evidence on the determinants of
offshore production provided by recent studies. Hanson, Mataloni,
and Slaughter (2005) show that U.S. multinational firms attract
larger shares of the sales of their foreign affiliates when the
latter benefit from lower trade costs and lower wages abroad. Kurz
(2006) shows that the U.S. plants and firms using imported
components in production are larger and more productive than their
domestically-oriented counterparts, as the larger productivity
allows them to cover the fixed costs of offshoring.

The study of offshoring motivated by lower production costs is
important to understand the macroeconomic interdependence between
country pairs and economic areas separated by persistent
differences in the cost of effective labor, such as the U.S. and
Latin America, or Western Europe and the new member countries of
the European Union (Marin, 2006; Meyer, 2006). In 2005, offshore
production through vertical integration was responsible for as much
as 50 percent of the manufacturing sales of the U.S. affiliates in
Mexico, and for 26 percent of the sales of the U.S. affiliates in
Latin America as a whole, shares which were directed towards the
U.S. parent firms (BEA, 2007).

The rest of this paper is organized as follows: Section 2
introduces the model of offshoring with heterogeneous firms that
allows for fluctuations in offshoring along its extensive and
intensive margins. Section 3 translates the model with firm
heterogeneity into an equivalent framework with two representative
firms that produce domestically and offshore. Section 4 describes
the model calibration. Section 5 presents the results; it shows the
business cycle dynamics of offshoring in the presence of aggregate
productivity shocks, and also compares the empirical moments of
offshoring from the U.S. to Mexico with their model counterparts.
Section 6 concludes with a summary and possible extensions of the
model.

2 Model of Offshoring with
Heterogeneous Firms

2.1 Markets and Production
Strategies

The model consists of two economies, North and South. Each
economy includes one representative household, and also a continuum
of firms that are monopolistically competitive and heterogeneous in
labor productivity, with each firm producing a different variety of
goods. Every period, the existing firms choose the destination
market(s) that they serve and the location of production, as
follows:

All firms serve their domestic market. For this purpose, the
Northern firms produce their varieties using either domestic or
foreign labor. The use of foreign labor involves the establishment
of an offshore production plant, and offers the advantage of a
lower production cost. However, offshoring is subject to per-period
fixed and trade costs, as described below.12

Some firms from each economy also serve the foreign market.
They use exclusively domestic labor in production, and export their
varieties subject to a per-period fixed cost, as in Ghironi and
Melitz (2005).

Because the number of firms in each economy varies over time,
and also because the existing firms re-optimize their offshoring
and exporting strategies every period, the composition of
consumption baskets in each economy changes over time.

This section illustrates the mechanisms of domestic and offshore
production as alternative choices for the Northern firms that
produce for their domestic market. Every period, the firm with
idiosyncratic labor productivity must choose one of
the two possible production strategies:

(a) Domestic production, with output being a function of the
aggregate productivity in the North , the
firm-specific labor productivity , and domestic labor
:

The Northern firm producing offshore becomes subject to the
Southern aggregate labor productivity but is
able to carry its own idiosyncratic labor productivity to the South.13

The monopolistically-competitive firm with idiosyncratic
productivity solves the profit-maximization problem
for the alternative scenarios of domestic and offshore
production:

(3)

(4)

where
and
are the prices associated with
each of the two production strategies, and
are the real wages in the North
and the South, and is the real exchange
rate. The cost of producing one unit of output either domestically
or offshore varies not only with the cost of effective labor
and
across
countries, but also with the level of idiosyncratic labor
productivity across firms.14 In addition, the
Northern firms producing offshore incur a fixed offshoring cost
equal to units of Southern effective
labor15 - that reflects the building and
maintenance of the offshore production facility - and also an
iceberg trade cost associated with the
shipping of goods produced offshore back to the parent country. For
every units produced offshore, only one unit
arrives in the North for consumption, as the difference is lost due
to trade barriers, transportation and insurance costs (Anderson and
Wincoop, 2004). The demand functions for the variety produced by
firm domestically or offshore are
and
,
where is the aggregate consumption in the
North.

The profit-maximization problem implies the equilibrium prices
and
for the alternative scenarios of domestic and offshore production.
The corresponding profits, expressed in units of the aggregate
consumption basket are:

(5)

(6)

When deciding upon the location of production every period, the
firm with productivity compares the profit
that it would obtain from domestic
production with the profit
from producing the same variety
offshore. As a particular case, I define the productivity cutoff
level on the support interval
, so that the firm at the
cutoff obtains equal profits from producing domestically or
offshore:

(7)

The productivity cutoff is a variable that
reacts to fluctuations in the relative cost of effective labor
across countries, and thus reflects the behavior of the extensive
margin of offshoring over the business cycle.

The model implies that only the relatively more productive
Northern firms find it profitable to produce their varieties
offshore. Despite the lower cost of effective labor in the South,
only firms with idiosyncratic productivity above a certain cutoff
() obtain benefits from offshoring
that are large enough to cover the fixed and iceberg trade costs.
This feature of the model is consistent with the empirical evidence
in Kurz (2006), that the U.S. plants and firms using imported
components in production are larger and more productive than their
domestically-oriented counterparts, as the larger idiosyncratic
productivity levels allow them to cover the fixed costs of
offshoring.16

Existence of the equilibrium
productivity cutoff Next I show that the existence of the equilibrium productivity
cutoff requires a cross-country asymmetry in
the cost of effective labor, so that some of the Northern firms
will always have an incentive to produce offshore. I begin by
re-writing the per-period profits that would be obtained from
domestic and offshore production as
and
, where
measures the size of the Northern market. In Figure 1, I plot the
two profits as functions of the idiosyncratic productivity
parameter
over the support interval
. The vertical intercept is
zero for the case of domestic production; it is equal to the
negative of the fixed cost in the case of offshoring (
).

Figure 1: The Firm specific productivity cutoff

The existence of the equilibrium productivity cutoff in Figure 1 requires that the profit function from
offshoring must be steeper than the profit from domestic
production,
When this condition is met, offshoring generates greater profits
than domestic production for the subset of firms with idiosyncratic
productivity along the upper range of the support
interval (). The slope inequality is
equivalent to
which implies that the effective wage in the South must be
sufficiently lower than that in the North, so that the difference
covers both the fixed cost and the iceberg trade cost (), and thus provides an incentive for some of the
Northern firms to produce offshore. The model calibration and the
magnitude of macroeconomic shocks ensure that this condition is
satisfied every period.17

2.3 Exporting Firms

Firms from each economy can choose to serve the foreign market
through exports as in Ghironi and Melitz (2005), in addition to
producing for their domestic market. In the North, the firm with
idiosyncratic productivity would use an amount
of domestic labor to produce for the
Southern market,
.18 The Southern
firms that choose to export to the North face a similar
problem.

Profit maximization implies the following equilibrium price and
profit functions for the Northern exporter with productivity factor
:
and
, where
is the aggregate consumption in
the South. Producing for the foreign market generates additional
profits, but involves a fixed exporting cost equal to units of Northern effective labor, and also an iceberg
trade cost
. Thus, the model implies that
only the subset of Northern firms with idiosyncratic labor
productivity above a productivity cutoff
find it profitable to export to the Southern market, as they can
afford the fixed and iceberg trade costs of exporting. The
time-varying productivity cutoff is:

(8)

2.4 Households and Consumption
Baskets

The representative household in the North maximizes the expected
lifetime utility, which is
subject to the budget constraint:

(9)

where is the amount of aggregate consumption,
is the subjective discount
factor, and is the inverse of the
inter-temporal elasticity of substitution. (The representative
household in the South faces a similar problem.) The Northern
household starts every period with share holdings in a mutual fund of firms whose
average market value is
, and also with real bond
holdings . It receives dividends equal to the
average firm profit
in proportion with the
number of firms and with its share holdings, interest
on bond holdings, and the real
wage for the amount of labor supplied inelastically.

The Northern household purchases two types of assets every
period. First, it purchases shares in a
mutual fund of Northern firms that includes: (i) firms already producing at time , either
domestically or offshore, and (ii) new
firms that enter the market in period . Each share
is worth its market value
, equal to the net present
value of the expected stream of future profits of the average firm.
Second, the household also purchases the risk-free bond denominated in units of the Northern consumption
basket.19 In addition, the household purchases
the consumption basket , which includes
varieties produced by the Northern firms either domestically
or offshore
; it also includes
varieties produced by Southern firms and imported by the North
:

(10)

where is the symmetric elasticity of
substitution across varieties. I use the home consumption basket
as the numeraire good, so that the
price index in the North is
, where
, and
is the real price of each
variety expressed in units of the Northern consumption basket. The
first-order conditions generate the following Euler equations for
bonds and stocks:

(11)

(12)

where is the exogenous rate of firm exit
every period.

2.5 Firm Entry and Exit

Firm entry takes place in both countries every period, as in
Ghironi and Melitz (2005). An unbounded pool of potential entrant
firms face a trade-off between the sunk entry cost (reflecting
headquarter activities in the parent country, such as research and
development, management, marketing) and the expected stream of
future monopolistic profits (discounted by the probability of exit
very period). In the North, firm entry requires a sunk entry cost
equal to units of Northern effective
labor.20 After paying the sunk entry cost,
each firm is randomly assigned an idiosyncratic labor productivity
factor that is drawn independently from a common
distribution with support over the interval
, and which the firm keeps
for the entire duration of its life.

The firms entering at time do not produce until period .
Irrespective of their idiosyncratic productivity, all firms -
including the new entrants - are subject to a random exit shock
that occurs with probability at the end of
every period. Thus, the law of motion for the number of producing
firms is
where
is the total number
of Northern firms that produce either domestically or offshore at
period .

The potential entrant firms anticipate their expected post-entry
value
, which depends on the
expected stream of future profits
, the stochastic discount
factor, and the exogenous probability of exit
every period. The forward iteration of the Euler equation for
stocks (12) generates the
following expression for the expected post-entry value of the
average firm:

(13)

In equilibrium, firm entry takes place until the expected value of
the average firm is equal to the sunk entry cost expressed in units
of the Northern consumption basket:

(14)

3 Aggregation over Heterogeneous
Firms

This section translates the model with firm heterogeneity into
an equivalent framework with two representative Northern firms that
produce domestically and offshore for their domestic market. Since
offshoring takes place one-way, there is only one representative
Southern firm that produces for the domestic market. In addition to
the firm producing for the domestic market, one representative firm
in each economy produces domestically for the export market.

3.1 Average Firm Productivity
Levels

Domestic vs. offshore
production First I describe the average productivity levels of the two
representative Northern firms that produce domestically and
offshore for the Northern market. In Figure 2, I plot the density
of the firm-specific labor productivity levels
over the support interval
. Every period , there are firms from the North
with idiosyncratic productivity levels below the offshoring cutoff
() that produce domestically;
their average productivity is
. There are also
firms with productivity factors
above the cutoff () that choose to
produce offshore; their average productivity is
.21 Since the
firm-specific labor productivity levels are random
draws from a common distribution with density
, I compute the two average productivity
levels as:

and

(15)

Figure 2: Average labor productivity levels of
Northern firms that produce domestically () and offshore () for the Northern market.

I assume that the firm-specific labor productivity draws
are Pareto-distributed, with p.d.f.
and c.d.f.
over the support
interval
. Using this assumption, I
derive analytical solutions for the average productivity levels of
the two representative Northern firms that produce domestically and
offshore as functions of the time-variant productivity cutoff
:22

and

(16)

where the productivity cutoff is
, and
the parameters are
and
.23 Since offshoring
takes place one-way, from the North to the South, the Southern
firms serve their domestic market exclusively through domestic
production. Their average productivity is constant,
, as it covers
the entire support interval
.

Exporting firms Under the assumption of Pareto-distributed productivity draws,
the average productivity levels of the exporting firms in each
economy are as in Ghironi and Melitz (2005):

and

(17)

3.2 Average Prices and Profits

Using the average productivity levels for the domestic,
offshoring and exporting firms described above, I translate the
model of offshoring in terms of three representative Northern
firms: one produces domestically, another produces offshore
(each serving the Northern market), while a third firm produces
domestically and exports to the Southern market. There are only
two representative Southern firms: one produces for the
local market, and the other exports to the North.

I describe the average prices and profits for each representative
firm in Table 1.

Using the property that the Northern firm at the productivity
cutoff is indifferent between the two
production strategies, I derive the following relationship between
the average profits of the two representative firms that produce
domestically and offshore:24

(18)

In addition, using the property that the firm at the
productivity cutoff obtains zero profits
from exporting, the average profits from exports are:

and

(19)

Table 1: Average Prices and Profits

Firm

Origin

Production

Market

Average Price

Average Profit

1.

North

North

North

2.

South

South

South

3.

North

South

North

4.

North

North

South

5.

South

South

North

Table 1 summarizes the average prices and profits of the firms that produce domestically for their local markets (rows 1 and 2); of the Northern firms that produce offshore (row 3); and of the firms that produce domestically for the export markets (rows 4 and 5).

Price indexes The consumption price index in the Northern economy is a
function of the average prices of varieties produced domestically
and offshore by the Northern firms, as well as the average price of
the varieties imported from the South:

(20)

In the South, there is no representative firm producing offshore.
The consumption price index depends on the average price of
varieties produced domestically by the Southern firms, and also on
that of varieties imported from the North:

(21)

Total profits The total profits of the Northern firms include the average
profits from domestic and offshore production, as well as those
from exports:

(22)

The total profits of the Southern firms combine the average profits
from domestic sales and exports:

(23)

3.3 Aggregate Accounting and the Current
Account Balance

I measure the aggregate income as the sum of the wage bill and
the amount of stock dividends that households in each economy
obtain every period,
and
The value added offshore, defined as the wage bill of the Southern
workers employed for production in the offshoring sector, is part
of the Southern output. The profits of the Northern firms producing
offshore are part of the Northern output.

Under financial autarky in the markets for bonds and stocks,
aggregate accounting implies that households spend their income
from labor and stock holdings on consumption and investment in new
firms,
and

The current account in the North is:

Under financial autarky, the balanced current account condition
() states that the sum of (a) exports
by Northern firms to the South and (b) repatriated profits of
offshore affiliates must be equal to the sum of (c) the value added
by offshore affiliates imported in the North and (d) the imports of
varieties produced by the Southern firms.25

3.4 Model Summary

The baseline model with financial autarky for the Northern
economy is characterized by 16 equations in 16 endogenous
variables: , ,
, , ,
,
,
,
,
,
,
,
, ,
and . Since the
Southern firms do not produce in the high-cost North, the Southern
economy is described by only 11 equations in 11 endogenous
variables; there are no Southern counterparts for , ,
,
and
. In particular, the
average labor productivity of the representative Southern firm
producing for the domestic market (
is constant over
time. Variables , ,
and
are predetermined.26

4 Calibration

I use a standard quarterly calibration by setting the subjective
rate of time discount
to match an average annualized
interest rate of 4 percent. The coefficient of relative risk
aversion is . Following Ghironi and Melitz
(2005), I set the intra-temporal elasticity of substitution
, and the probability of firm exit
to match the annual 10 percent
job destruction in the U.S.

As summarized in Table 2, I calibrate the fixed costs of
offshoring () and exporting ( and
) as well as the Pareto
distribution parameter , so that the model matches
the importance of offshoring and trade for the Mexican economy in
steady state, as illustrated by four empirical moments: (1) The
maquiladora value added represents about 20 percent of Mexico's
manufacturing GDP (INEGI, 2008), compared to 25 percent in the
model; (2) The maquiladora exports represent about half of Mexico's
total exports (Bergin, Feenstra, and Hanson, 2008), vs. 60 percent
in the model; (3) Employment in the maquiladora sector accounts for
approximately 25 percent of Mexico's total manufacturing employment
(Bergin, Feenstra, and Hanson, 2008), compared to 22 percent in the
model; (4) Total imports represent the equivalent of 33 percent of
Mexico's GDP (INEGI, 2008), and 32 percent in the model. To this
end, I set
(the fixed cost of offshoring
for Northern firms),
and
(the fixed costs of
exporting for the Northern and Southern firms, respectively), as
well as (the Pareto distribution
coefficient).27 Without loss of generality, I set
the lower bound of the support interval for firm-specific
productivity in the North and the South at
.

Table 2 summarizes the calibration of key parameters of the model, and also compares the steady state values of key macroeconomic variables with their empirical counterparts.

In order to obtain a steady-state asymmetry in the cost of
effective labor across countries, I set the sunk entry cost - which
reflects headquarter activities sensitive to the regulation of
starting a business in the firms' country of origin - to be larger
in the South than in the North (
and ). As a result, the steady state number of firms, the
labor demand and the real wage are relatively lower in the South.
The calibration reflects the considerable variation in the cost of
starting a business across countries: The corresponding monetary
cost is 3.3 times higher in Mexico than in the U.S. or Canada; it
is 6.2 times higher in Hungary than in the U.K. (World Bank, 2007).
The asymmetric sunk entry costs, along with the iceberg trade cost
and the values for , and
reported above, generate a
steady state value for the terms of labor that is less than one (
).28 In
other words, the steady state cost of effective labor in the South,
defined as real wage divided by aggregate productivity, is
76 percent of the corresponding level in the
North. The calibration provides an incentive for the Northern firms
to produce offshore in steady state.

The resulting steady-state fraction of the Northern firms that
use foreign labor () is 1.4 percent; the fraction of exporting firms () is 10.1 percent. Since I model
offshoring in an asymmetric two-country framework that abstracts
from the exchanges of U.S. firms with the rest of the world (other
than Mexico), the steady state values reported above are less than
their empirical counterparts. In the data, approximately 14 percent
of the U.S. firms (other than domestic wholesalers) used imported
inputs from both Mexico and the rest of the world in 1997 (Bernard,
Jensen, Redding and Schott, 2007). Out of that, intra-firm imports
represented half of the total amount, while the rest was accounted
by arm's length transactions (Bardhan and Jafee, 2004).29Approximately 21 percent of the U.S.
manufacturing plants were exporters in 1992 (Bernard, Eaton, Jensen
and Kortum, 2003).

The calibration also implies that, in the North, the
steady-state expenditure share of the varieties produced by
Northern firms domestically (66.0 percent) - firms which are
relatively less productive than the average - is less than their
fraction in the total number of varieties available in the North
(89.2 percent). In contrast, since the offshored varieties are
produced by the relatively more productive Northern firms, their
expenditure share (21.2 percent) is more than their fraction in the
total number of varieties available in the North (1.2 percent). The
pattern is consistent with the more productive firms having larger
market shares than their less productive counterparts.30

5 Results

5.1 Offshoring to Mexico's Maquiladora
Sector

In this section I describe the cyclicality of offshoring
motivated by lower production costs using data from U.S.
manufacturing and Mexico's maquiladora sector. In particular, I
document the business cycle dynamics of the extensive margin of
offshoring from the U.S. to Mexico, which will be useful to assess
the implications of the model with offshoring that are described in
the following sections.

Mexico's maquiladora sector The maquiladora sector represents an appropriate empirical setup
to study the cyclicality of offshoring by U.S. manufacturing firms
motivated by lower production costs, due to the absence of local
consumption in Mexico and its direct links to U.S. manufacturing.
The plants operating under Mexico's maquiladora program import
inputs, process them, and ship the resulting goods back to the
country of origin (Gruben, 2001).31 Although only a
subset of the maquiladora plants are U.S.-owned, most of them
accommodate the offshoring operations of U.S. firms: They import
most of their inputs from the U.S. (82 percent), and export most of
their output (90 percent) back to the U.S. (Hausman and Kaytko,
2003; Burstein, Kurz and Tesar, 2009). The value added of the
maquiladora sector is part of Mexico's manufacturing output.

Empirical cross-correlations Mexico's total manufacturing output and, in particular, the
maquiladora value added are strongly correlated with U.S.
manufacturing. In Figure 3 (panel 1), I plot the detrended series
for Mexico's maquiladora value added (the dashed line) and Mexico's
total manufacturing output (the dotted line) against the
manufacturing component of U.S. industrial production (henceforth
U.S. IP, the solid line), for the interval between 1990:Q1 and
2006:Q4.32 The chart shows that the U.S.
recessions in 1990 and 2001, as well as the expansion throughout
the late 1990s, were associated with similar patterns in the
maquiladora value added. During the 1994-95 financial crisis in
Mexico, the decline in the maquiladora value added was less
pronounced than the drop in Mexico's total manufacturing output, as
the offshoring sector in Mexico benefited from the direct links
with U.S. manufacturing. The cross-correlations in panel 2 show
that Mexico's maquiladora value added moves closely together with
the U.S. manufacturing output, and that its correlation with U.S.
manufacturing is larger than that of Mexico's total manufacturing
output.

In panel 3 (bottom left), I plot the detrended series for the
number of maquiladora plants in Mexico (the dashed line) - which
reflects the extensive margin of offshoring - against the U.S. IP
for manufacturing (the solid line). The cross-correlations in panel
4 show that U.S. manufacturing leads the number of maquiladora
plants by about four quarters. The result suggests that the
extensive margin of offshoring adjusts gradually over time, whereas
the maquiladora value added is contemporaneously correlated with
the U.S. manufacturing output.

Figure 3: Business cycles dynamics of offshoring to Mexico

5.2 Impulse Responses of
Offshoring

I log-linearize the baseline model of offshore production under
financial autarky around the steady state, and compute the impulse
responses to a transitory one-percent increase in aggregate
productivity in the North. I assume that aggregate productivity is
described by the autoregressive process
, with the
persistence parameter .

Figure 4 shows the impulse responses of the baseline model of
offshoring (thick solid lines), and contrasts them to those from
two alternative frameworks: (i) a model of offshoring in which the
productivity cutoff is fixed, so that the fraction of offshoring
firms is constant over the business cycle (thin solid
lines)33; and (ii) the extreme case with no
offshoring, a case in which I replicate the model with exports in
Ghironi and Melitz (2005) (dashed lines). For each variable, the
horizontal axis illustrates quarters after the initial shock, and
the vertical axis shows the percent deviations from the original
steady state in each quarter.

The intensive margin In the baseline model (thick solid lines), on impact, the
increase in aggregate labor productivity in the North generates a
proportional increase in the real wage . The
rising demand for varieties produced both domestically and offshore
causes an immediate increase in offshoring along its intensive
margin (i.e. offshore value added per firm). Since the increase in
aggregate productivity in the North is not replicated in the South,
the excess demand for Southern effective labor causes the real wage
in the South
and the terms of
labor
to jump. As a result, the number of Northern firms that produce
offshore () drops on impact due to: (i) the
increase in the cost of effective labor offshore, and (ii) the
increase in the fixed cost of offshoring, both of which are
sensitive to the effective wage in the South.

The extensive margin As the aggregate labor productivity in the North persists above
its initial steady state, the larger market size encourages firm
entry, as shown by the gradual increase in the number of incumbent
firms (). In turn, firm entry leads to an
increase in the demand for Northern labor, which causes the cost of
effective labor to appreciate gradually in the North relative to
the South. In Figure 4, this appreciation is visible as the real
wage in the North declines more slowly after the initial shock than
aggregate productivity, and thus the terms of labor persist below
their initial steady state level. Following the appreciation of the
terms of labor, the number of offshoring firms () increases, as some of the more productive Northern
firms relocate production to the South. Notably, the increase in
the number of offshoring firms is gradual, as it mirrors the
gradual appreciation of the terms of labor.

The total value added offshore ()
increases by more under the baseline model of offshoring (thick
solid line) than in the alternative model of offshoring in which
the productivity cutoff is fixed (thin solid line). Thus, 20
quarters after the shock, more than half of the increase in the
total value added offshore is due to the adjustment along the
extensive margin.

In the South, the initial jump in the real wage - caused by the
spike of the intensive margin of offshoring - is followed by an
additional increase which occurs gradually over time, as some of
the more productive Northern firms relocate production to the
South. Since the increase in offshoring along its extensive margin
transfers some of the upward pressure from the domestic to the
foreign wage, the terms of labor appreciate by less ( declines by less) in the baseline model of offshoring
(thick solid line) than in the alternative models with no extensive
margin adjustment (thin solid line) and no offshoring (dashed
line).

5.3 The Extensive and Intensive Margins
of Offshoring

In this section I provide evidence in support of the model of
offshoring with extensive margin adjustments over the business
cycle. Thus, I analyze the empirical cross-correlations between
lags and leads of the U.S. manufacturing output and two empirical
indicators of offshore production in Mexico: (i) the number of
maquiladora establishments, as an empirical proxy for the extensive
margin; and (ii) the value added per establishment, as a proxy for
the intensive margin. Then I compare the empirical correlations to
their model counterparts.

In the model, the total value added offshore, VA
, is a function of the number of offshoring firms, their average
idiosyncratic productivity, the foreign cost of effective labor,
and the aggregate consumption in the North. The number of
offshoring firms () measures the
extensive margin of offshore production, and constitutes the
counterpart of the number of maquiladora plants in Mexico. The real
value added per offshoring firm (VA
) represents the intensive
margin of offshoring; it is the model counterpart of the value
added per maquiladora plant.34

Model vs. empirical cross-correlations Figure 5 (panels 1 and 2) shows the empirical correlations
between U.S. manufacturing and the two margins of the maquiladora
sector (black line), together with their 95 percent confidence
intervals. It also shows the model correlations generated by the
baseline model of offshoring under financial autarky (red solid
line), as well as those generated by the baseline model augmented
with elastic labor supply (green dotted line). Aggregate
productivity follows the bivariate autoregressive process:

(24)

where the persistence parameters are
, and the
spillovers are
, as
in Backus, Kehoe, and Kydland (1992). The variance of the shocks is
and the covariance is
, values which correspond
to a correlation of innovations of 0.258.

Regarding the extensive margin (panel 1), the data shows a
strong and positive correlation between the number of maquiladora
plants and the past U.S. manufacturing output. As discussed,
expansions in U.S. manufacturing tend to lead the number of
offshore plants by about four quarters. The model is successful in
capturing this pattern; the correlation between the number of
offshoring firms and the past output in the North is
positive, and reaches a peak for the Northern output lagged by four
quarters. The result is explained by the fact that, following a
productivity improvement in the North, the increase in the number
of offshoring firms is gradual, as it mirrors the gradual
appreciation of the terms of labor caused by domestic firm entry.
Although the contemporaneous correlation between the number of
offshoring firms and Northern output is slightly negative35
(rather than positive as in the data), the model replicates the
inter-temporal dynamics of the extensive margin.

Turning towards the intensive margin (panel 2), the empirical
correlation between the maquiladora value added per plant and the
past U.S. manufacturing output is negative and statistically
significant. The model is successful in replicating this pattern as
well. Following a positive technology shock in the North, the
number of offshoring firms increases faster than the total value
added offshore due to the appreciation of the terms of labor. As a
result, the value added per offshoring firm declines below its
initial level several quarters after the shock, and the correlation
between the intensive margin of offshoring and past output in the
North is negative.

5.4 The Co-Movement of Output and
Offshore Production

In this section I illustrate the cross-country correlations of
Northern output and the value added offshore generated by the
baseline model of offshoring, and examine their sensitivity to the
trade cost and the persistence of aggregate productivity.

The co-movement of output and the
value added offshore I assume that aggregate productivity follows the bivariate
autoregressive process described by equation (24). The
persistence parameters are asymmetric across the two economies (
and
), there are no
spillovers (
), and
the technology shocks are less volatile in the North than in the
South (i.e. variances
vs.
), with the covariance
implying a correlation of
shocks of 0.27. These assumptions are based on the
estimates of the bivariate productivity process for the U.S. and
Mexico in Mandelman and Zlate (2008), that use data on total factor
productivity (TFP) for the two countries.

Table 3 shows the cross-country correlation of output
, the correlation
of output in the country of origin with the value added offshore
, and also the
cross-country correlation of consumption
generated by the
model of offshoring.36 It also reports the correlations
generated by the alternative framework in which I shut down
offshoring, and thus replicate the model with endogenous exports in
Ghironi and Melitz (2005).

Table 3. Cross-country contemporaneous
correlations

Model:

Financial autarky: Offshoring

Financial autarky: No offshoring

International Bond trading: Offshoring

International Bond trading: No offshoring

Corr(YR,Y*R)

0.35

0.27

0.34

0.32

Corr(YR,VAR)

0.99

n/a

0.99

n/a

Corr(CR,C*R)

0.40

0.28

0.44

0.37

Table 3 summarizes the cross-country contemporaneous correlations of output, offshore value added and consumption generated by the models with and without offshoring, for the cases of financial autarky (left panel) and financial integration (right panel).

Two notable results emerge from Table 3. First, the correlation
of Northern output with the value added offshore is larger than the
cross-country correlation of total output,
, a
result which is consistent with the empirical correlations
documented above (i.e. the maquiladora value added co-moves more
closely with the U.S. manufacturing output than does Mexico's total
manufacturing output.) In the model, the value added offshore is
closely related to aggregate productivity, net firm entry and the
appreciation of the cost of effective labor in the North, and thus
is strongly correlated with the Northern output. In contrast, the
total output in the South, on one hand, receives the positive
contribution of the offshoring sector, which enhances its
co-movement with the Northern output. On the other hand, the
relative productivity increase in the North dampens firm entry in
the South, and thus partially offsets the additional co-movement
generated by offshoring. In addition, offshoring reduces the
competitiveness of the Southern exporters, as the relocation of
production transfers some of the upward pressure from the Northern
wage to the Southern one, which further dampens output co-movement.
Nonetheless, under international bond trading, the cross-country
co-movement of total output is further reduced by the
resource-shifting effect that occurs as households lend across the
border to finance firm entry in the country receiving the favorable
productivity shock.

Second, the model with offshoring enhances the co-movement of
output relative to the special case with no offshoring.
Intuitively, in the model with no offshoring in Ghironi and Melitz
(2005), the positive shock to aggregate productivity in the North
generates an increase in the demand for all varieties - produced
both in the North and in the South - as well as an immediate
increase in the relative price of Southern varieties. The resulting
substitution away from Southern varieties is offset by the increase
in net firm entry and the gradual appreciation of the cost of
effective labor in the North, which enhances the export
competitiveness of the Southern firms. Therefore, the model with no
offshoring still generates positive output co-movement across the
two economies.

Offshoring introduces an additional transmission mechanism that
enhances the co-movement of output. Following an increase in
aggregate productivity in the North, firm entry causes a gradual
appreciation of the terms of labor, which in turn provides an
incentive for some of the more productive Northern firms to
relocate production to the South. The increase of output in the
North (generated by the positive shock to aggregate productivity)
and also in the South (generated by the initial jump in Northern
demand for offshored varieties and, subsequently, the gradual
relocation of production by the Northern firms to the South),
enhance the co-movement of total output across the two economies.
The effect is dampened under international trade in bonds, as the
Southern households lend to finance firm entry during expansions in
the North.

The remainder of this section examines the sensitivity of
cross-country correlations to: (a) variation in the iceberg trade
cost and (b) variation in the persistence of
the bivariate autoregressive productivity process . Figures 6 and 7 show that the model with offshoring
generates larger cross-country correlations for both output and
consumption relative to the special case with no offshoring,
results which hold for a wide range of possible values for the
trade cost
and
the persistence parameter
.

Sensitivity to the trade cost
The cross-country correlation of output is greater for lower
values of the trade cost (Figure 6). During expansions in the
North, a lower trade cost (that applies to both offshoring and non
offshoring-related trade) enhances the demand for varieties
produced in the South (either by Northern offshoring firms or by
Southern exporters). A lower trade cost also facilitates the
relocation of production offshore over the business cycle, and thus
enhances the cross-country co-movement of output.

Figure 6: Cross-country correlations of output and consumption, sensitivity to

The result is consistent with the empirical regularity
documented in Burstein, Kurz, and Tesar (BKT, 2009), namely that
country pairs with (i) larger shares of offshoring-related trade in
bilateral trade and (ii) larger bilateral trade flows relative to
output also exhibit larger correlations of manufacturing output. In
BKT (2009), the regression of output correlations between the U.S.
and foreign economies on (i) the production-sharing intensity of
foreign exports and (ii) the share of exports in foreign output
generates OLS coefficients that are positive and statistically
significant (0.746 and 0.140,
respectively). In the model considered in this paper, a decline in
the trade cost from its baseline calibration value to the lower
extreme (i.e. from 1.3 to 1.21) is associated with an 11 percentage
point increase in the correlation of output (Figure 6, panel 1 for
financial autarky). The same reduction in the trade cost is linked
to (i) an increase in the steady-state share of offshoring-related
trade in the Southern exports (from 60 to 92 percent), and also
(ii) an increase in the share of exports in the Southern output
(from 40 to 86 percent). The resulting slope between the output
correlation and the steady-state share of offshoring-related trade
in Southern exports (0.344) is roughly half the
corresponding OLS coefficient in BKT (2009). The slope between the
output correlation and the steady-state share of exports in
Southern output (0.240) is slightly larger than its
empirical counterpart.

Sensitivity to the aggregate
productivity persistence The model with offshoring generates larger cross-country
correlations of output than the alternative model with no
offshoring for the entire range of values of the persistence
parameter .37 More, the additional
co-movement generated by offshoring increases with the persistence
parameter. Under financial autarky (panel 1 in Figure 7), the
additional correlation brought by offshoring increases from
6 to 12 percentage points
as the persistence parameter rises from
to
. Following a positive
technology shock in the North, the larger persistence of aggregate
productivity leads to a larger increase in domestic demand, and
thus to a larger increase in offshoring along its intensive margin
on impact. The larger persistence also enhances firm entry and
generates a larger appreciation of the terms of labor over the
business cycle, which in turn provides a greater incentive for
firms to relocate production offshore. Under financial integration
(panel 2), the result is dampened by the resource-shifting effect
that occurs when households lend to the country that receives a
positive technology shock.

Figure 7: Cross-country correlations, sensitivity to

5.5 Real Exchange Rate Dynamics

Offshoring narrows the price dispersion across countries, as it
dampens the appreciation of the real exchange rate that follows a
domestic increase in aggregate productivity. In the absence of
offshoring, the framework with firm entry and endogenously traded
varieties in Ghironi and Melitz (2005) generates the
Harrod-Balassa-Samuelson effect (i.e. more productive economies
exhibit higher average prices), as the country that receives a
favorable shock to aggregate productivity also experiences an
appreciation of the terms of labor and a rise in import prices.
However, offshoring dampens this effect though a number of
channels, including the transfer of upward pressure from the
domestic to the foreign wage, the reduced size of the domestic
non-traded sector, and the decline in import prices as offshoring
crowds out the less productive foreign exporters.

Average prices and product
variety I use the consumer price index (CPI)-based real exchange rate
as the theoretical counterpart for the empirical real exchange
rate, since the average prices
and
best represent the
corresponding empirical CPI levels in the presence of endogenous
product variety (Broda and Weinstein, 2003). To this end, I break
down the welfare-based price indexes and
into components reflecting (a)
product variety and (b) average prices as
for the North and
for the South. The resulting expression for the CPI-based real
exchange rate is:

(25)

where the terms of labor
measures the cost of effective labor in the South relative to the
North; the iceberg trade costs and
affect the imports of the
North and the South, respectively.

Analytical results The log-linearized version of (25) is:

(C1)

(C2)

(C3)

(C4)

(C5)

where the variables marked with a hat denote percent deviations
from their steady states. Parameter
is the steady-state share of
spending in the North on varieties produced by Northern firms
both domestically and offshore;
is the steady-state share of
spending in the North only on varieties produced by Northern
firms offshore (I shut down offshoring when
and
);
is the steady-state share of
spending in the South on varieties produced by Southern firms. The
calibration ensures that: (a)
,
as the domestically-produced varieties represent more than 50
percent of consumption spending in each country; (b)
and
i.e. the market shares of varieties produced domestically by the
less productive firms are smaller than their fraction
in the total number of varieties; and (c)
, i.e. the market share of varieties produced offshore by
themore productive Northern firms is larger than
their fraction in the total number of varieties available in the
North. The model implies that the relatively more productive
offshoring firms have larger market shares than their less
productive domestic counterparts, which is in line with the
empirical evidence in Kurz (2006).

The log-linearized form of (25) outlines five
channels (labeled C1-C5 in the
log-linearized equation above) through which the CPI-based real
exchange rate is affected by: (1) changes in the price of
non-traded varieties induced by fluctuations in the terms of labor
(
); (2) changes in the price
of offshored varieties caused by fluctuations in the average
productivity of offshoring firms
and in the magnitude of trade costs
; (3)
changes in relative import prices triggered by fluctuations in the
average productivity of Southern exporters
relative to that of their Northern counterparts
; (4) changes
in the number of varieties produced by the Northern firms offshore
relative to the
number of imported varieties produced by Southern firms (
); and (5) changes
in the number of domestic varieties (
) relative to the
number of imported Southern varieties (
).

Impulse responses Offshoring dampens the appreciation of the real exchange rate
that follows an increase in aggregate productivity in the North.
The effect occurs through channels C1, C3 and C4 defined above.
Figure 8 describes the effect of each channel on the real exchange
rate in the baseline model of offshoring with financial autarky: It
shows the impulse responses of the real exchangte rate and related
variables to a transitory one-percent increase in aggregate
productivity in the North, when productivity follows the
autoregressive process
with
persistence .

(C1) Changes in the price of non-traded varieties. In the
model with no offshoring (dashed lines), a productivity increase in
the North encourages firm entry and leads to the appreciation of
the terms of labor in the medium run (i.e.
decreases). This causes the average price of non-traded varieties
in the North to increase relative to that in the South, and thus
leads to the appreciation of the CPI-based real exchange rate (i.e.
decreases).

Offshoring (thick solid line) dampens the appreciation of the
real exchange rate through this channel in two ways: (a) Offshoring
dampens the appreciation of the terms of labor (i.e. causes
to decrease by less) relative to the
alternative models with no offshoring (dashed line) or to the model
with a fixed productivity cutoff (thin solid line), because the
relocation of production transfers upward pressure from the
domestic to the foreign wage. (b) Offshoring also reduces the
impact of the terms of labor on the real exchange rate, since it
reduces the share of non-traded varieties in total spending. The
effect is illustrated by the coefficient on
in channel C1 (
), which
decreases with the share of offshored varieties in total spending (
).

(C2) Changes in the price of offshored varieties. On
impact, the increase in the Southern wage causes the number of
offshoring firms to drop and their average productivity to
increase. However, offshoring becomes an increasingly profitable
option in the medium run due to the gradual appreciation of the
terms of labor. As a result, the number of offshoring firms rises,
their average productivity
declines, and their
average price increases over time. Thus, offshoring contributes to
the appreciation of the real exchange rate in the medium run
through this channel.38

(C3) Changes in relative import prices. In the absence of
offshoring, the appreciation of the terms of labor reduces the
export profitability of the Northern firms relative to that of
their Southern counterparts. As a result, the average productivity
of the surviving Northern exporters
increases relative to
that of Southern exporters
,
and their average price declines. This causes an increase in the
average price of imports in the North relative to the South, which
leads to the appreciation of the real exchange rate.

Offshoring reverses this effect. As the relocation of production
places upward pressure on the Southern wage, offshoring harms the
export competitiveness of Southern firms, and causes the average
productivity of the surviving Southern exporters
to
increase relative to that of their Northern counterparts
. In contrast to the
model with exports only, offshoring causes a decline in the average
price of imports in the North relative to the South, and therefore
dampens the appreciation of the real exchange rate.

(C4) Expenditure switching from imports towards offshored
varieties. As offshoring reduces the competitiveness of
Southern exports, consumers in the North switch their expenditure
away from the less competitive Southern varieties (
decreases) and towards the
relatively cheaper varieties produced offshore (
increases). The result dampens
the appreciation of the real exchange rate.

(C5) Expenditure switching from imports towards domestic
varieties. Firm entry in the North causes the number of
domestic varieties (
) to increase relative to that
of imported foreign varieties (
). Thus, consumers switch
their expenditure away from imported varieties and towards the
varieties produced domestically by the relatively less productive
firms, which are available at relatively higher prices. This
channel works towards the appreciation of the real exchange
rate.

Figure 8: Impulse responses to one-percent shock to aggregate productivity in the North

6 Conclusion

I study the effect of offshoring on the cross-country
transmission of business cycles, while focusing on its extensive
and intensive margins as separate transmission mechanisms. The
paper considers a model of offshoring with heterogeneous firms that
is consistent with the empirical patterns of offshoring from U.S.
manufacturing to Mexico's maquiladora sector. First, following an
aggregate productivity increase in the country of origin (North),
the value added per offshoring firm jumps on impact and then
returns to its initial steady state. However, domestic firm entry
causes a gradual increase in the relative cost of effective labor
(i.e. the wage adjusted by aggregate productivity), which in turn
generates a gradual increase in the number of offshoring firms (the
extensive margin), as in the data. Second, offshoring enhances the
cross-country co-movement of output relative to the model with
endogenous exports. The result is consistent with the empirical
regularity documented in Burstein, Kurz, and Tesar (2009) that
country pairs with larger shares of offshoring-related trade in
bilateral trade also exhibit larger correlations of manufacturing
output. Third, offshoring reduces the appreciation of the real
exchange rate that follows an aggregate productivity improvement in
the parent country, and thus dampens the Harrod-Balassa-Samuelson
effect that occurs in the framework with firm entry and
endogenously traded varieties.

There are a number of possible extensions to the model
considered in this paper. First, the framework is useful to analyze
the impact of offshore production on employment in the parent and
the host countries. Second, a possible extension with rich policy
implications would involve the study of interactions between
offshore production and labor migration in a unified framework, in
which both offshoring and labor mobility are driven by fluctuations
in the relative wage across countries. Third, while this paper
studies the fluctuations of offshoring over the business cycle,
further research should address the long-run developments in
offshore production and its implications for U.S.
manufacturing.

A.2 Offshoring with International Bond
Trading

I introduce international bond trading in the model with
offshoring. International asset markets are incomplete, as the
representative household in each economy holds risk-free,
country-specific bonds from both the North and the South. Each type
of bonds provides a real return denominated in units of the issuing
country's consumption basket. Quadratic costs of adjustment for
bond holdings ensure stationarity for the net foreign assets in the
presence of temporary shocks.

The representative household in the North maximizes
inter-temporal utility subject to:

(26)

where and
are the rates of return of the
North and South-specific bonds;
and
denote the
principal and interest income from each type of bonds;
and
are the
adjustment costs for each type of bond holdings; is the fee rebate. Setting , I
add the two Euler equations for bonds to the baseline model:

(27)

(28)

For the Southern representative household, the Euler equations for
bonds are:

(29)

(30)

The market clearing conditions for bonds are:

(31)

(32)

Thus, financial integration through trade in bonds adds four new
variables (
) and six new equations
(27, 28, 29, 30, 31 and 32) while removing
the original two Euler equations from the baseline model with
financial autarky. Also, the new expressions for aggregate
accounting in the North and the South are:

(33)

(34)

Finally, I replace the balanced current account condition from
financial autarky with the expression for the balance of
international payments:

(35)

which shows that the current account balance (trade balance plus
repatriated profits plus investment income) must equal the negative
of the financial account balance (the change in bond holdings).

Average Firm-Specific Productivity
Levels

Firms producing offshore I obtain the average productivity of the Northern firms that
produce offshore by integrating over the upper range of the support
interval, above the offshoring productivity cutoff:

(36)

(37)

where

Firms producing domestically The average productivity of the Northern firms that produce
domestically is:

(38)

(39)

A.4 Average Profits from Domestic and
Offshore Production

The average profit of the Northern firms producing domestically
is:

(40)

The average profit of the Northern firms producing offshore
is:

(41)

The Northern firm with productivity equal to the cutoff
is indifferent between producing
domestically or offshore. I use the equality of profits at the
productivity cutoff,
, and
equations (40) and
(41) to write the
link between the two average profits as:

(42)

A.5 The Real Exchange Rate

The CPI-based real exchange rate is
, in which
and
are the total number of varieties available to consumers in the
North and the South. For the price indexes and
, I use the average price
formulas implied by the broader framework of offshoring described
in footnotes 11 and 16 to obtain:

(43)

In what follows I use the notation
to denote the steady-state share of spending in the North on
varieties produced by the Northern firms both domestically and
offshore. Expression
denotes the steady-state share of spending in the North on goods
produced by the Northern firms offshore only. (Note that
.) Expression
denotes the steady-state share of spending in the South on goods
produced by the Southern firms domestically. The average
productivity of the Southern firms producing domestically
is constant over
time. Using all of the above, I log-linearize the CPI-based real
exchange rate:

(44)

I set
so that my model of
offshoring nests the framework with endogenous exports in Ghironi
and Melitz (2005): In addition to the firms that produce
domestically or offshore for their home market, a subset of firms
from each economy also serve the foreign markets through exports.
The log-linearized expression for the CPI-based real exchange rate
becomes:

(45)

A.6 Asymmetric Steady State

In this section I provide the steady state solution for the
model of offshoring in the presence of a cross-country asymmetry in
the cost of effective labor (i.e. the terms of labor ). To this end, I use the broad framework described
in footnotes 11 and 16 of the paper that nests both the baseline
model of offshoring (for
) and the model with exports
only in Ghironi and Melitz (2005) (for
).

I obtain a numerical solution for the unique steady state using
a non-linear system of 12 equations in 12 unknowns. The equations
are described by 46-57 below. The
unknowns are the steady state values of (the
offshoring productivity cutoff in the North), (the exporting productivity cutoff in the North),
(the terms of labor),
(the real consumption
ratio in units of the same consumption basket),
(the real exchange rate),
(the real profits from domestic and offshore production for the
domestic market, as well as the profit from production for the
export market, each divided by the real wage in the
North) (the exporting productivity
cutoff in the South),
(the average price of
Northern exports),
(the average
price of Southern exports), and
(the ratio of the
number of firms in the North and the South). Subsequently, I use
the numerical solutions for these 12 variables to compute the
steady state values for the remaining variables.

The following price and profit formulas (in which the aggregate
productivity is
) are useful in computing the
steady state solution:

Table A.2. - Panel A: Average Prices

1. Domestic production, North

2. Domestic production, South

3. Offshore production ()

4. Exports () or horiz. FDI (), North

5. Exports () or horiz. FDI (), South

Table A.2. - Panel B: Average Profits

1. Domestic production, North

2. Domestic production, South

3. Offshore production ()

4. Exports () or horiz. FDI (), North

5. Exports () or horiz. FDI (), South

Introducing
,
, and
in the expression for the total
profits in the North (see Table A.1), the first equation of the
system is:

Using the expression for the total profits in the South (see
Table A.1), it follows that:

(51)

The consumption ratio in units of the same consumption basket
is:

(52)

From the balanced current account condition, I obtain:

(53)

where
and

The expression for the real exchange rate in steady state
is:

(54)

The remaining equations are:

(55)

(56)

(57)

with:

A.7 Empirical Impulse Responses of
Offshoring to Mexico

I estimate the empirical impulse responses of offshoring to
Mexico's maquiladora sector (total value added, number of plants,
and the value added per plant) to permanent technology shocks in
U.S. manufacturing. To this end, I use a structural VAR model with
five variables: (i) labor productivity in U.S. manufacturing, (ii)
labor productivity in Mexico's maquiladora, (iii) value added per
plant and (iv) the number of plants in Mexico's maquiladora, as
well as (v) hours worked in U.S. manufacturing. The estimation
details are discussed in Zlate (2009). With the exception of the
intensive margin, all variables have a unit root and therefore
enter the VAR model in first differences. My identification
strategy assumes that long-run labor productivity in U.S.
manufacturing responds exclusively to U.S. technology shocks.
Conversely, long-run labor productivity in Mexico's maquiladora
sector - which uses production machinery received on loan from U.S.
firms - responds to both the U.S. and Mexico-specific permanent
technology shocks.

In Figure A.1, I plot the estimated impulse responses of
Mexico's maquiladora variables, together with the +/- 2 standard
error confidence intervals. Following a positive, one standard
deviation, permanent technology shock to U.S. manufacturing, the
number of maquiladora plants (the extensive margin) does not react
on impact, but increases gradually over time. The value added per
maquiladora plant (the intensive margin) exhibits an immediate
jump, followed by an additional increase until it reaches a peak
two quarters after the shock. The intensive margin then declines
below its initial level, but returns to it over time.

The predictions of the theoretical model of offshoring
(illustrated in Figure 4) are consistent with the empirical impulse
responses. In the model, following a positive transitory shock to
aggregate productivity in the North, the extensive margin of
offshoring increases gradually over time, despite the initial drop.
The intensive margin jumps on impact, then declines below its
initial steady state and returns to it in the medium run, as in the
data.

Figure A.1: Empirical impulse responses of offshore production to a permanent U.S. technology shock.

Footnotes

1. I am grateful to Fabio Ghironi, James
Anderson and Susanto Basu for their help during my dissertation
work at Boston College. Special thanks to George Alessandria,
Richard Arnott, David Arseneau, Marianne Baxter, Paul Bergin,
Michele Cavallo, Bora Durdu, Matteo Iacoviello, Peter Ireland,
Federico Mandelman, Joel Rodrigue, Vitaliy Strohush, Linda Tesar
and Christina Wang for insightful discussions on this paper. I
would like to recognize participants at the 2010 Winter Meeting of
the Econometric Society, the 2009 SCIEA Meeting of the Federal
Reserve, the IEFS/ASSA 2009 Meeting, the 4th Dynare Conference at
the Boston Fed, the Green Line Macro Workshop at Boston
College/Boston University (Spring 2008), as well as the R@BC and
Dissertation Workshops at Boston College for important suggestions.
Dissertation fellowships at the FRB of Atlanta and the FRB of
Boston were extremely useful for the development of this
paper. Return to text

2. Contact: Andrei.Zlate@frb.gov, (202)
452-3542. The author is a staff economist in the Division of
International Finance, Board of Governors of the Federal Reserve
System, Washington, D.C. 20551 U.S.A. The views in this paper are
solely the responsibility of the author and should not be
interpreted as reflecting the views of the Board of Governors of
the Federal Reserve System or of any other person associated with
the Federal Reserve System. Return to
text

3. "Offshoring" refers to the activity of
firms that relocate certain stages of production to foreign
countries; firms can either become integrated across borders
through vertical or/and horizontal FDI, or purchase intermediate
goods and services from unaffiliated foreign suppliers. In
contrast, "outsourcing" applies to firms that purchase
intermediates from unaffiliated suppliers - either at home or
abroad - rather than producing them in house (see Helpman,
2006). Return to text

5. Bergin, Feenstra and Hanson (2008) show
that the extensive margin of offshoring accounts for more than one
third of the adjustment of industry-level employment, and for
nearly half of the adjustment of total employment in Mexico's
maquiladora sector. Return to
text

7. In Burstein, Kurtz and Tesar (2008), the
low elasticity of substitution between the domestic and foreign
goods enhances the cross-country co-movement of
output. Return to text

8. I maintain a one-to-one correspondence
between an offshoring firm, a variety, and an offshore
plant. Return to text

9. In the traditional international real
business cycle literature, in contrast, a domestic increase in
aggregate productivity leads to increased production at home but
not offshore, as in Backus, Kehoe and Kydland (1992). Return to text

10. Recent empirical literature highlights
the role of the extensive margin in international trade in the
presence of fixed exporting costs: Baldwin and Harrigan (2007) show
that the number of traded goods (the extensive margin) decreases
with distance and increases with the size of the importing country.
Besedes and Prusa (2006) find that the survival rate of exports for
differentiated good varieties increases with the initial
transaction size and also with the length of the relationship.
Hummels and Klenow (2005) show that larger economies have larger
exports, and that the extensive margin accounts for as much as 60
percent of this difference. Return to
text

11. "Exporter hysteresis" refers to the
behavior of firms that continue to serve the foreign market even
after a real exchange rate appreciation reduces their export
competitiveness. Return to text

12. All Southern firms produce domestically
because the higher cost of effective labor in the North offers them
no incentive to produce offshore. Return
to text

13. Strategies (1) and (2) are the special
cases of a more general framework of offshoring, in which the
offshoring firm with idiosyncratic labor productivity uses a combination of Northern and Southern labor,
and
. The output of firm is a Cobb-Douglas function of domestic and foreign inputs,
, as in Antras and Helpman (2004). In this paper, I explore two
special cases: At one extreme, I set to
shut down offshore production, a case which replicates Ghironi and
Melitz (2005). At the other extreme, I set
so that the firms choosing to produce offshore use exclusively
foreign inputs. The smaller , the larger
the range of operations that the offshoring firms relocate abroad.
I use the l'Hôpital rule and
to obtain:
Return to text

14. The cost of effective labor is the
ratio between the real wage and aggregate productivity in each
country. The real exchange rate
is the ratio
between the price indexes in the South and the North expressed in
the same currency, where
is the nominal exchange
rate. Return to text

15. The cost of
units of Southern effective labor is equivalent to
units of the
Southern consumption basket. Return to
text

16. A useful implication of model with
firm heterogeneity is that the more productive firms have larger
output and revenue. Given two firms with idiodsyncratic
productivity
, the ratios of output and
profits are
and
(see Melitz, 2003). Empirical studies show that firms using
imported inputs in production are not only more productive, but
also have larger revenues and employ more workers (Kurz,
2006). Return to text

17. A second condion necessary to avoid
the corner solution when all firms would produce offshore is that
. It
ensures that
in all
periods. Return to text

18. I view exporting as a special case
within a more general framework, in which firms serve the foreign
market using a mix of domestic and foreign inputs in production:
where a larger is equivalent to a smaller
content of Southern inputs used in the production of goods sold in
the South. In my model, I incorporate the special case with
endogenous exports as in Ghironi and Melitz (2005) by setting
. Alternatively, I would model the case
in which Northern firms serve the Southern market exclusively
through their foreign affiliates (as in Contessi, 2006) by setting
. Return to
text

19. In the model with financial autarky
(in which mutual fund shares and bonds are not traded across
countries), the equilibrium conditions for stock and bond holdings
are
and
. Bond holdings play a role
in the extended model with incomplete financial markets, in which
the representative household buys risk-free, country-specific bonds
in the presence of quadratic adjustment costs for bond holdings
(Appendix A.2). Return to text

20. The sunk entry cost is equivalent to
units of the Northern
consumption basket. Return to
text

21. Note that
is the average
productivity of offshoring firms, whereas
is the productivity cutoff above which firms produce
offshore. Return to text

23. I use the Pareto c.d.f.
and the
share of Northern firms producing offshore
to
write the productivity cutoff as
. The
share of Northern firms producing domestically is
.
Parameter reflects the dispersion of the
productivity draws: A relatively larger implies a
smaller dispersion and a higher concentration of productivities
towards the lower productivity bound
. Return
to text

25. In the case with international trade
in bonds, the current account balance equals the change in bond
holdings, which is the negative of the financial account balance
(Appendix A.2). Return to text

26. The model summary is in Appendix A.1.
The steady-state solution is available in Appendix
A.6. Return to text

27. In the alternative model with exports
only, I set
and
so that the fraction of
Northern exporting firms (10 percent) and that of Southern
exporting firms (63 percent) match the corresponding steady state
values from the model with offshoring. Return to text

28. The terms of labor is the ratio
between the cost of effective labor in the South and the North
expressed in units of the same consumption basket. The calibration
ensures that the condition
from Section 2.2 is satisfied
in steady state. Return to text

29. The value of 14 percent would
understate the fraction of plants that use foreign inputs if
the offshoring firms tend to operate multiple plants that
produce different varieties. Return to
text

30. In the South, the share of Southern
varieties in total spending (61.7 percent) is less than their
fraction in the total number of varieties (62.8 percent), since
Northern exporters are more productive than the average Southern
firm. Return to text

32. The seasonally adjusted data in
natural logs is expressed in deviations from a Hodrick-Prescott
trend. The data for U.S. manufacturing IP is provided by the
Federal Reserve Board. The data for Mexico's manufacturing IP and
the maquiladora sector (real value added and the number of plants)
is provided by INEGI (2008). I aggregate the maquiladora data into
quarters (from the original monthly frequency), and seasonally
adjust it using the X-12-ARIMA method of the U.S. Census
Bureau. Return to text

33. In the alternative model with fixed
productivity cutoff, the fraction of offshoring firms is constant,
but the number of offshoring firms varies over time due to firm
entry in the country of origin. During expansions in the North, the
new entrants that draw idiosyncratic productivity factors above the
cutoff start by producing directly offshore. However, none of the
firms that initially produce at home can relocate offshore when the
terms of labor appreciate. Return to
text

34. I deflate the value added offshore by
the average price index of the varieties produced offshore, VA
VA
. To this end, I
decompose the price index for the offshore varieties into
components reflecting (a) variety and (b) average price as
, to obtain VA
VA. Return to text

35. Following a positive shock to
aggregate productivity in the North, the initial drop in the number
of offshoring firms - caused by a spike in the Southern wage - is
followed by a gradual increase above the initial steady state
level, as the terms of labor appreciate over time. Return to text

36. In order to compute the cross-country
correlations, I deflate output and consumption by the average price
indexes in each country, since the empirical price deflators are
best represented by the average price index
rather than the
welfare-based price index . For instance,
I use
to deflate output in the North as
I deflate the value added offshore by the average price index of
the varieties produced offshore as VA
VA
VA. Return to
text

37. I assume that the persistence
parameter is symmetric across the North and the South, that there
are zero spillovers, but maintain the variance-covariance matrix of
shocks from Mandelman and Zlate (2008). Return to text

38. Exogenous policy changes can also
affect the price of goods produced offshore. For instance, tariff
cuts for the varieties produced offshore, reflected by a decrease
in , would dampen the appreciation of
the CPI-based real exchange rate. Return
to text